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Abstract
It is known that every contact form on a closed three-manifold has at least two
simple Reeb orbits, and a generic contact form has infinitely many. We show that if
there are exactly two simple Reeb orbits, then the contact form is nondegenerate.
Combined with a previous result, this implies that the three-manifold is
diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are
the core circles of a genus-one Heegaard splitting. We also obtain further information
about the Reeb dynamics and the contact structure. For example, the Reeb flow
has a disk-like global surface of section and so its dynamics are described
by a pseudorotation, the contact structure is universally tight, and in the
case of the three-sphere the contact volume and the periods and rotation
numbers of the simple Reeb orbits satisfy the same relations as for an irrational
ellipsoid.
Keywords
Reeb orbit, embedded contact homology, lens space,
pseudorotation
Mathematical Subject Classification
Primary: 37J99, 53E50
Secondary: 53D42
Publication
Received: 29 September 2021
Revised: 24 January 2022
Accepted: 26 February 2022
Published: 5 December 2023
Proposed: András I Stipsicz
Seconded: Ciprian Manolescu, Leonid Polterovich
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